000110594 001__ 110594 000110594 005__ 20240319081002.0 000110594 0247_ $$2doi$$a10.1007/s00220-021-04284-8 000110594 0248_ $$2sideral$$a127164 000110594 037__ $$aART-2022-127164 000110594 041__ $$aeng 000110594 100__ $$aCedzich C. 000110594 245__ $$aQuantum Walks: Schur Functions Meet Symmetry Protected Topological Phases 000110594 260__ $$c2022 000110594 5060_ $$aAccess copy available to the general public$$fUnrestricted 000110594 5203_ $$aThis paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points ± 1. The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples. 000110594 536__ $$9info:eu-repo/grantAgreement/ES/DGA/E26-17R$$9info:eu-repo/grantAgreement/ES/DGA/E48-20R$$9info:eu-repo/grantAgreement/EC/FP7/337603/EU/Multipartite Quantum Information Theory/QMULT$$9info:eu-repo/grantAgreement/EC/FP7/600645/EU/Simulators and Interfaces with Quantum Systems/SIQS$$9info:eu-repo/grantAgreement/ES/MICINN-AEI-FEDER/MTM2017-89941-P 000110594 540__ $$9info:eu-repo/semantics/openAccess$$aby$$uhttp://creativecommons.org/licenses/by/3.0/es/ 000110594 590__ $$a2.4$$b2022 000110594 592__ $$a1.413$$b2022 000110594 591__ $$aPHYSICS, MATHEMATICAL$$b11 / 56 = 0.196$$c2022$$dQ1$$eT1 000110594 593__ $$aStatistical and Nonlinear Physics$$c2022$$dQ1 000110594 593__ $$aMathematical Physics$$c2022$$dQ1 000110594 594__ $$a4.3$$b2022 000110594 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion 000110594 700__ $$aGeib T. 000110594 700__ $$aGrünbaum F.A. 000110594 700__ $$0(orcid)0000-0002-3050-9540$$aVelázquez Campoy, L. F.$$uUniversidad de Zaragoza 000110594 700__ $$aWerner A.H. 000110594 700__ $$aWerner R.F. 000110594 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada 000110594 773__ $$g389 (2022), 31-74$$pCommun. Math. Phys.$$tCommunications in Mathematical Physics$$x0010-3616 000110594 8564_ $$s799833$$uhttps://zaguan.unizar.es/record/110594/files/texto_completo.pdf$$yVersión publicada 000110594 8564_ $$s1751544$$uhttps://zaguan.unizar.es/record/110594/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada 000110594 909CO $$ooai:zaguan.unizar.es:110594$$particulos$$pdriver 000110594 951__ $$a2024-03-18-14:14:19 000110594 980__ $$aARTICLE